PostScript is a page description language, a protocol that is used to communicate between applications (like Adobe InDesign, QuarkXPress or MS Word) and output devices, like laser printers or platesetters, that are equipped with a PostScript interpreter. PostScript can also be used to exchange data between applications themselves and it forms the basis of the PDF file format.

Now I can do some relations:
Postscrip level 3, for exemple, it is a progression in the language and PostScript level 3 RIPs mean that RIP can interpret this level of language. The way it make this interpretation, I think it is: or by CPSI (I suppose that it is a cheaper OEM tecnology, than APPE) or by APPE. I worked with Prinergy and saw Normalizer process occur in front of me all the time.
After a little research (certainly not something new for the Sr., or as we say here in Brazil: teaching the priest to pray!), CPSI v3015 had some problem with transparency, solved with flattened PDF 1.3, nowadays v3019 without thoses issues.

PostScript is a page description language. A PostScript file defines what a page or document looks like.

CPSI and APPE are software libraries that Adobe sells to vendors so these can build a RIP (or renderer as it is sometimes called). Such a RIP takes a page description and uses it to output the document, usually on physical media such as paper, film or printing plates.

Let me use an analogy: A PostScript file is to an MP3 file as a RIP is to a music application on your computer or the software of your MP3 player.

How do CPSI and APPE relate to PostScript:

CPSI is the precursor of APPE. It is a PostScript interpreter: you can send a PostScript file to a RIP based on CPSI and it will make sure you get a nice print-out, or a good set of films or plates.

APPE was created when Adobe had already phased out PostScript. APPE cannot process PostScript files directly, it expects to be fed PDF files. A lot (or maybe all) of the vendors that build RIPs based on APPE still see the need for supporting PostScript so they typically add a PostScript to PDF convertor (Distiller or Normalizer) to their RIP or workflow. That means you can still send PostScript files to such a system. In the background they will be converted to PDF and then forwarded to the APPE component.

one question. You wrote: “PostScript can also be used to exchange data between applications themselves and it forms the basis of the PDF file format.” I am not clear about it. Could you clarify? Or can I understand in this way:if I export a PDF from indesign, and I use different PDF viewer to read it, such as Preview, Acrobat or VPS, I may get different display result because they have different RIP built in?